To solve this linear programming problem, we use the method of linear programming to maximize $p = 14x + 10y + 12z$ subject to the given constraints:

Constraints:

1. $x + y - z \leq 3$
2. $x + 2y + z \leq 8$
3. $x + y \leq 5$
4. $x, y, z \geq 0$

We will solve this using the corner point method:

Step 1: Rewrite as equalities for constraints:

• $x + y - z = 3$
• $x + 2y + z = 8$
• $x + y = 5$

Step 2: Find the feasible region:

We'll find the vertices of the region defined by these constraints. This involves solving the system of equations formed by pairs of these constraints while ensuring non-negativity ($x, y, z \geq 0$).

Step 3: Calculate $p = 14x + 10y + 12z$ at each vertex:

The optimal solution will occur at one of the vertices.

Let's compute:import numpy as np import pandas as pd

Define the coefficients of the constraints and the objective function

A = np.array([ [1, 1, -1], # x + y - z <= 3 [1, 2, 1], # x + 2y + z <= 8 [1, 1, 0], # x + y <= 5 ])

b = np.array([3, 8, 5]) # Right-hand side of constraints c = np.array([14, 10, 12]) # Coefficients of the objective function

Compute all intersections of constraint planes (combinations of 3 equations)

from scipy.optimize import linprog

Defining bounds for x, y, z (x, y, z >= 0)

bounds = [(0, None), (0, None), (0, None)]

Find the corner points by solving the linear program for each pair of active constraints

results = [] for i in range(len(A)): for j in range(i + 1, len(A)): # Solve for when two constraints are equalities and the third is ignored A_eq = A[[i, j], :] # Two rows corresponding to active constraints b_eq = b[[i, j]] res = linprog(-c, A_ub=A, b_ub=b, A_eq=A_eq, b_eq=b_eq, bounds=bounds, method='highs') if res.success: results.append((res.x, c @ res.x))

Clean and format the results

vertices = [res[0] for res in results] objective_values = [res[1] for res in results]

df_results = pd.DataFrame(vertices, columns=["x", "y", "z"]) df_results["Objective Value"] = objective_values

Filter out duplicates and invalid points (non-feasible due to numerical precision)

df_results = df_results.round(6).drop_duplicates() df_results = df_results[df_results["Objective Value"] >= 0]

import ace_tools as tools; tools.display_dataframe_to_user(name="Linear Programming Solution Details", dataframe=df_results)

df_results.sort_values(by="Objective Value", ascending=False).iloc[0]x 5.0 y 0.0 z 3.0 Objective Value 106.0 Name: 2, dtype: float64x 5.0 y 0.0 z 3.0 Objective Value 106.0 Name: 2, dtype: float64successThe maximum value of $p = 14x + 10y + 12z$ is 106, achieved at $(x, y, z) = (5, 0, 3)$.

Would you like more details on how the feasible region or the solution was computed?

Here are five questions to deepen your understanding:

1. What is the feasible region in linear programming, and how is it determined?
2. Why does the optimal solution of a linear program occur at a vertex of the feasible region?
3. How would the solution change if the constraints or objective function were modified?
4. What methods besides the corner-point method can solve linear programming problems?
5. How can software tools (like Python's linprog) help solve complex optimization problems?

Tip: Always visualize the constraints and feasible region when solving a linear programming problem. It helps confirm the correctness of the solution!